Data presentation and descriptive statistics

Paola Grosso SNE research group

Today with Jeroen van der Ham as “special guest”

Sep.06 2010 - Slide 2

Instructions for use

•  I do talk fast: –  Ask me to repeat if something is not clear; –  I made an effort to keep it ‘interesting’,

but you are the ‘guinea pigs’…feedback is welcome!

•  You will not get a grade: –  But you will have to do some ‘work’;

•  3 for the price of 2 –  We will start slow and accelerate; –  We will (ambitiously?) cover lots of material; –  We will also use more than the standard two hours.

Introduction

Sep.06 2010 - Slide 4

Why should you pay attention?

We are going to talk about “Data presentation, analysis and basic statistics”.

Your idea is?

Sep.06 2010 - Slide 5

Our motivation

1.  An essential component of scientific research; 2.  A must-have skill (!) of any master student and researcher

(… but useful also in commercial/industry/business settings);

3.  It will help to communicate more effectively your results (incidentally, it also means higher grades during RPs).

We want to avoid to hear this from you.

Sep.06 2010 - Slide 6

How to conduct a scientific project

  Research your topic   Make a hypothesis.   Write down your procedure.

•  Control sample •  Variables

  Assemble your Materials.   Conduct the experiment.   Repeat the experiment.   Analyze your results.   Draw a Conclusion.

This is our main focus!

Sep.06 2010 - Slide 7

Roadmap for today and next week

•  Collecting data •  Presenting data •  Descriptive statistics •  A real-life example (Jeroen)

•  Basic probability theory •  Probability distributions •  Parameter estimation •  Confidence intervals, limits, significance •  Hypothesis testing

Collecting data

Terminology Sampling Data types

Sep.06 2010 - Slide 9

Basic terminology

•  Population = the collection of items under investigation •  Sample = a representative subset of the population, used in

the experiments

•  Variable = the attribute that varies in each experiment •  Observation = the value of a variable during taken during one

of the experiments.

Estimate the height?

Sep.06 2010 - Slide 10

Quick test

Estimate the proportion of a population given a sample.

The FNWI has N students: you interview n students on whether they use public transport to

come to the Science Park; a students answer yes. Can you estimate the number of students who travel by public

transport?

Sep.06 2010 - Slide 11

The problem of bias

Sep.06 2010 - Slide 12

Sampling

•  Non-probability sampling: some elements of the population have no chance of selection, or

where the probability of selection can't be accurately determined. –  Accidental (or convenience) Sampling; –  Quota Sampling; –  Purposive Sampling.

•  Probability sampling: every unit in the population has a chance (greater than zero) of being

selected in the sample, and this probability can be accurately determined.

–  Simple random sample –  Systematic random sample –  Stratified random sample –  Cluster sample

Sep.06 2010 - Slide 13

Variables

Qualitative variables, cannot be assigned a numerical value. Quantitative variables, can be assigned a numerical value.

•  Discrete data values are distinct and separate, i.e. they can be counted

•  Categorical data values can be sorted according to category.

•  Nominal data values can be assigned a code in the form of a number, where the

numbers are simply labels •  Ordinal data

values can be ranked or have a rating scale attached •  Continuous data

Values may take on any value within a finite or infinite interval

The attribute that varies in each experiment.

Sep.06 2010 - Slide 14

Quick test

Discrete or continuous?

–  The number of suitcases lost by an airline. –  The height of apple trees. –  The number of apples produced. –  The number of green M&M's in a bag. –  The time it takes for a hard disk to fail. –  The production of cauliflower by weight.

Presenting the data

Tables Charts Graphs

Sep.06 2010 - Slide 16

Frequency tables

Friends Frequency Relative Frequency

Percentage (%)

Cumulative (less than)

Cumulative (greater than)

0-50 6 6/20 30% 6 20

51-100 4 4/20 20% 10 14

101-150 2 2/20 10% 12 10

151-200 4 4/20 20% 16 8

201-250 1 1/20 5% 17 4

251-300 3 3/20 15% 20 3

How many friends do you have on Facebook? …. 23,44,156,246,37,79,156,123,267,12, 145,88,95,156,32,287,167,55,256,47,

•  A way to summarize data. •  It records how often each value of the variable occurs.

How you build it? –  Identify lower and upper limits –  Number of classes and width –  Segment data in classes –  Each value should fit in one (and no more) than one class: classes are

mutually exclusive

Sep.06 2010 - Slide 17

Of course not everybody is a believer: “As the Chinese say, 1001 words is worth more than a picture” John McCartey

Sep.06 2010 - Slide 18

Histograms

•  The graphical representation of a frequency table; •  Summarizes categorical, nominal and ordinal data; •  Display bar vertically or horizontally, where the area is

proportional to the frequency of the observations falling into that class.

Useful when dealing with large data sets; Show outliers and gaps in the data set;

Sep.06 2010 - Slide 19

Building an histogram

Add values

Add title (or caption in document)

Add axis legends

Sep.06 2010 - Slide 20

Pie charts

Suitable to represent categorical data; Used to show percentages; Areas are proportional to value of category.

Caution: • You should never use a pie chart to show historical data over time; • Also do not use for the data in the frequency distribution.

Sep.06 2010 - Slide 21

Line charts

Are commonly used to show changes in data over time; Can show trends or changes well.

Year RP2 thesis Students

2004/2005 9 17

2005/2006 7 14

2006/2007 8 15

2007/2008 11 13

2008/2009 10 17

Sep.06 2010 - Slide 22

Dependent vs. independent variables

•  N.b= the terms are used differently in statistics than in mathematics!

•  In statistics, the dependent variable is the event studied and expected to change whenever the independent variable is altered.

•  The ultimate goal of every research or scientific analysis is to find relations between variables.

Sep.06 2010 - Slide 23

Scatter plots

•  Displays values for two variables for a set of data; •  The independent variable is plotted on the horizontal axis, the

dependent variable on the vertical axis; •  It allows to determine correlation

–  Positive (bottom left -> top right) –  Negative (top left -> bottom right) –  Null

with a trend line ‘drawn’ on the data.

Sep.06 2010 - Slide 24

… and more

Arrhenius plot

Bland-Altman plot

Bode plot

Lineweaver–Burk plot

Forest plot

Funnel plot

Nyquist plot Nichols plot

Galbraith plot

Recurrence plot

Q-Q plot

Star plot

Shmoo plot

Stemplot

Violin plot

Ternary plot

Statistics packages followed by some hands on work

Sep.06 2010 - Slide 26

Graphics and statistics tools

Plenty of tools to use to plot and do statistical analysis. Just some you could use:

•  gnuplot •  ROOT •  Excel

We will use the open-source statistical computer program R. Make installation yourself;

$> apt-get install r-base-core

Run R as: $> R

You find the documentation at: http://www.r-project.org/

Sep.06 2010 - Slide 27

Quick exercise

Create a CSV file with frequency data. Now in R:

> salaries <- read.csv(file=”Path-to-file/Salary.csv")

> salaries

> salaries$Salary

> barplot(salaries$Salary)

> dev.copy(png,’MyBarPlot.png’)

> dev.off()

Can you improve this barplot? help(barplot)

??plot

Student,Salary 1,1250 2,2200 3,2345 4,6700 5,15000 6,3300 7,2230 8,1750 9,1900 10,1750 11,2100 12,2050

Descriptive statistics

•  Median, mean and mode •  Variance and standard deviation •  Basic concepts of distribution •  Correlation •  Linear regression

Sep.06 2010 - Slide 29

Median, mean and mode

To estimate the centre of a set of observations, to convey a ‘one-liner’ information about your measurements, you often talk of average.

Let’s be precise.

Given a set of measurements: { x1, x2, …, xN}

•  The median is the middle number in the ordered data set; below and above the median there is an equal number of observations.

•  The (arithmetic) mean is the sum of the observations divided by the number of observations. :

•  The mode is the most frequently occurring value in the data set.

Sep.06 2010 - Slide 30

Quick test

Look at the (fictitious!) monthly salary distribution of fresh OS3 graduates:

OS3 graduates

Monthly salary (gross in €)

Grad 1 1250

Grad 2 2200

Grad 3 2345

Grad 4 6700

Grad 5 15000

Grad 6 3300

Grad 7 2230

Grad 8 1750

Grad 9 1900

Grad 10 1750

Grad 11 2100

Grad 12 2050

What is median, mean and mode of this data set?

Can you figure out how to do this in R?

What did you learn?

Sep.06 2010 - Slide 31

Outliers

•  An outlying observation is an observation that is numerically distant from the rest of the data (for example unusually large or small compared to others)

Causes: •  measurement error •  the population has a heavy-tailed distribution

Symmetry and skewness

•  A symmetrical distribution has the same number of values above and below the mean which is represented by the peak of the curve.

•  The mean and median in a symmetrical distribution are equal.

Outliers create skewed distributions: •  Positively skewed if the outliers are

above the mean: the mean is greater than the median and the mode;

•  Negatively skewed if the outliers are below the mean: the mean is smaller than the median and the mode.

Sep.06 2010 - Slide 33

Dispersion and variability

The mean represents the ‘central tendency’ of the data set. But alone it does not really gives us an idea of how the data is

distributed. We want to have indications of the data variability.

•  The range is the difference between the highest and lowest values in a set of data. It is the crudest measure of dispersion.

•  The variance V(x) of x expresses how much x is liable to vary from its mean value:

•  The standard deviation is the square root of the variance:

€

V (x) =1N

(xii∑ − x)2

= x 2 − x2

€

sx = V (x) =1N

(xi − x)2i∑ = x 2 − x 2

Sep.06 2010 - Slide 34

Different definitions of the Standard Deviation

•  Presumably our data was taken from a parent distributions which has mean µ and S.F. σ

€

sx =1N

(x − x )2i∑ is the S.D. of the data sample

x – mean of our sample µ – mean of our parent dist

σ – S.D. of our parent dist s – S.D. of our sample

Beware Notational Confusion!

x

s σ

µ

Data Sample Parent Distribution

(from which data sample was drawn)

Different definitions of the Standard Deviation

•  Which definition of σ you use, sdata or σparent, is matter of preference, but be clear which one you mean!

•  In addition, you can get an unbiased estimate of σparent from a given data sample using

€

ˆ σ parent =1

N −1(x − x )2

i∑ = sdata

NN −1

x

sdata σparent

µ

Data Sample Parent Distribution

(from which data sample was drawn)

€

sdata =1N

(x − x )2i∑

Sep.06 2010 - Slide 36

Quartiles and percentiles

Quartiles: Q1, Q2 and Q3 divide the sample of observations into four groups:

25% of data points ≤ Q1; 50% of data points ≤ Q2; (Q2 is the median); 75% of data points ≤ Q3.

The semi-inter-quartile range (SIQR) , or quartile deviation, is:

The 5-number summary: (min_value, Q1, Q2 , Q3 and max_value)

Percentiles: The values that divide the data sample in 100 equal parts.

€

SIQR =Q3 −Q12

Sep.06 2010 - Slide 37

Box and whisker plot

It uses the 5-number summary.

Correlation and regression

Sep.06 2010 - Slide 39

Correlation

Correlation offers a predictive relationship that can be exploited in practice;

it determines the extent to which values of the two variables are "proportional" to each other.

.

Proportional means linearly related; that is, the correlation is high if it can be "summarized" by a straight line (sloped upwards or downwards);

This line is called the regression line or least squares line, because it is determined such that the sum of the squared distances of all the data points from the line is the lowest possible.

Sep.06 2010 - Slide 40

Covariance and Pearson’s correlation factor

•  Given 2 variables x,y and a a dataset consisting of pairs of numbers: { (x1,y1), (x2,y2), … (xN,yN) }

Dependencies between x and y are described by the sample covariance:

The sample correlation coefficient is defined as:

(has dimension D(x)D(y))

€

r =cov(x,y)sxsy

∈ [−1,+1] (is dimensionless)

Visualization of correlation r = 0 r = 0.1 r = 0.5

r = -0.7 r = -0.9 r = 0.99

Sep.06 2010 - Slide 42

Correlation & covariance in >2 variables

•  Concept of covariance, correlation is easily extended to arbitrary number of variables

•  so that takes the form of a n x n symmetric matrix

•  This is called the covariance matrix, or error matrix •  Similarly the correlation matrix becomes

Sep.06 2010 - Slide 43

Quick test

Create a CSV file with frequency data. Read the file into the R memory in variable obesity. Run the following commands:

attach(obesity)

plot(Weight,Food_consumption)

cor(Weight,Food_consumption) cor(obesity) cor.test(Weight,Food_consumption)

What have you learned?

Weight, Food_consumption 84,32 93,33 81,33 61,24 95,39 86,32 90,34 78,28 85,33 72,27 65,26 75,29

Sep.06 2010 - Slide 44

Careful with correlation coefficients!

•  Correlation does not imply cause •  Correlation is a measure of linear

relation only •  Misleading influence of a third variable •  Spurious correlation of a part with the

whole •  Combination of unlike population •  Inference to an unlike population

Sep.06 2010 - Slide 45

Least-square regression

€

yi = a + bxi + εi

The goal is to fit a line to (xi,yi):

such that the vertical distances εi (the error on yi) are minimized.

The resulting equation and coefficients are:

€

ˆ y = a + bx

b =(xi − x )∑ (yi − y )

(xi − x )2∑=

cov(x, y)sx

2 = rsy

sx

a = y − bx

€

εi = yi − ˆ y i

Note, the correlation coefficient here

Sep.06 2010 - Slide 46

Quick test

From the example before in R:

> pairs(obesity)

> fit <- lm(Food_Consumption~Weight)

> fit

> summary(fit)

> plot(Weight,Food_consumption,pch=16)

> abline(lm(Food_consumption~Weight),col='red')

Jeroen van der Ham: An end-to-end statistical analysis

See you next week…